poly-m ers.
Let us de ne a function P (l1;k1;l2;k2;t) as the probability of nding the two-stranded polym erin thecon guration (l1;k1;l2;k2).H ereli;ki= 0;1;:::
(ki li,i = 1 or 2) are two independent param eters that count the total num ber of subunits (li) and the num ber of unhydrolyzed subunits (ki) in the i-th proto lam ent. W e assum e that the polym erization and hydrolysis in the actin lam ent can be described by \one-layer" approach (20,22). It m eans that l2= l1 or l2= l1+ 1,and k2 = k1 or k2 = k1 1 (see Fig. 1).
T hen the probabilities can be described by a set ofm aster equations. For con gurations w ith l1= l2= land 1 k < lwe have
dP(l;k;l;k;t)
dt = uP (l 1;k 1;l;k;t)+ wTP(l;k;l+ 1;k + 1;t)
+ rhP(l;k + 1;l;k;t) (u + wT + rh)P (l;k;l;k;t); (A 1)
and
dP(l;k;l;k 1;t)
dt = uP (l 1;k 1;l;k 1;t)+ wTP(l;k;l+ 1;k;t) + rhP(l;k;l;k;t) (u + wT + rh)P (l;k;l;k 1;t):(A 2)
Sim ilarly for the con gurations w ith l1= l2 1 = land 1 k < l+ 1 the m aster equations are
dP(l;k 1;l+ 1;k;t)
dt = uP (l;k 1;l;k 1;t)+ wTP(l+ 1;k;l+ 1;k;t) + rhP(l;k;l+ 1;k;t) (u + wT + rh)P (l;k 1;l+ 1;k;t);(A 3)
and
dP(l;k;l+ 1;k;t)
dt = uP (l;k;l;k 1;t)+ wTP(l+ 1;k + 1;l+ 1;k;t) + rhP(l;k;l+ 1;k + 1;t) (u + wT + rh)P (l;k;l+ 1;k;t):(A 4)
T hen the polym ercon gurationsw ithoutAT P-actin m onom ers(k = 0)can be described by
dP(l;0;l;0;t)
dt = wTP(l;0;l+ 1;1;t)+ wDP(l;0;l+ 1;0;t)
(u + wD)P (l;0;l;0;t); (A 5)
and
dP(l;0;l+ 1;0;t)
dt = wTP(l+ 1;1;l+ 1;0;t)+ wDP(l+ 1;0;l+ 1;0;t) (u + wD)P (l;0;l+ 1;0;t): (A 6)
Finally,for the con gurations consisting ofonly unhydrolyzed subunits we
T he conservation ofprobability leads to
+ 1X
Follow ing the m ethod of D errida (40), we de ne two sets of auxiliary
functions (k = 0;1;:::),
N ote that the conservation ofprobability gives us
X+ 1 k= 0
Bk;k0 (t)+ Bk+ 1;k0 (t)+ Bk;k+ 11 (t)+ Bk;k1 (t)+ B0(t)+ B1(t)= 1: (A 12)
T hen from the m aster equations (A 1),(A 2),(A 3) and (A 4) we derive for
k 1
dB0k;k(t)
dt = uBk1 1;k(t)+ wTBk;k+ 11 (t)+ rhBk+ 1;k0 (t) (u + wT + rh)Bk;k0 (t);
dBk1 1;k(t)
dt = uBk0 1;k 1(t)+ wTB0k;k(t)+ rhBk;k1 (t) (u + wT + rh)Bk1 1;k(t);
dBk;k0 1(t)
dt = uBk1 1;k 1(t)+ wTB1k;k(t)+ rhBk;k0 (t) (u + wT + rh)Bk;k0 1(t);
dB1k;k(t)
dt = uBk;k0 1(t)+ wTBk+ 1;k0 (t)+ rhBk;k+ 11 (t) (u + wT + rh)Bk;k1 (A 13)(t);
w hile the m aster equations (A 5),(A 6) for k = 0 yield
dB0;00 (t)
dt = wTB01;1(t)+ wDB10;0(t)+ rhB10;0(t) (u + wD)B00;0(t);
dB01;0(t)
dt = wTB10;0(t)+ wDB00;0(t)+ rhB01;1(t) (u + wD)B01;0(t):(A 14) Finally,equations (A 7) and (A 8) lead to
dB0(t)
dt = (u + wT)B1(t) (u + wT + rh)B0(t);
dB1(t)
dt = (u + wT)B0(t) (u + wT + rh)B1(t); (A 15) Sim ilar argum ents can be used to describe the functions Ck;k0 , Ck+ 1;k0 ,
Ck;k+ 11 ,and Ck;k1 . Speci cally,for k 1 we obtain
dCk;k0 (t)
dt = uCk1 1;k(t)+ wTCk;k+ 11 (t)+ rhCk+ 1;k0 (t) (u + wT + rh)Ck;k0 (t) +12[uBk1 1;k(t) wTB1k;k+ 1(t)]; (A 16)
dCk1 1;k(t)
dt = uCk0 1;k 1(t)+ wTCk;k0 (t)+ rhCk;k1 (t) (u + wT + rh)Ck1 1;k(t) +12[uBk0 1;k 1(t) wTBk;k0 (t)]; (A 17)
dCk;k0 1(t)
dt = uCk1 1;k 1(t)+ wTCk;k1 (t)+ rhCk;k0 (t) (u + wT + rh)Ck;k0 1(t) +12[uBk1 1;k 1(t) wTBk;k1 (t)]; (A 18)
dCk;k1 (t)
dt = uCk;k0 1(t)+ wTCk+ 1;k0 (t)+ rhCk;k+ 11 (t) (u + wT + rh)Ck;k1 (t) +12[uBk;k0 1(t) wTB0k+ 1;k(t)]: (A 19)
For k = 0 the expressions are the follow ing,
dC00;0(t)
dt = wTC01;1(t)+ wDC01;0(t)+ rhC10;0(t) (u + wD)C00;0(t)
1
2[wTB01;1(t)+ wDB01;0(t)]; (A 20)
dC01;0(t)
dt = wTC10;0(t)+ wDC00;0(t)+ rhC01;1(t) (u + wD)C01;0(t)
1
2[wTB10;0(t)+ wDB00;0(t)]: (A 21)
A gain follow ing the D errida’sapproach (40)we introduce an ansatz that should be valid at large tim es t,nam ely,
Bk;mi (t)! bik;m; Ck;mi (t)! aik;mt+ Tk;mi (i= 0;1;jk m j 1): (A 22)
A t steady state dBk;mi (t)=dt= 0,and Eqs.A 13 and A 14 yield for k 1
0 = ub1k 1;k+ wTb1k;k+ 1+ rhb0k+ 1;k (u + wT + rh)b0k;k; 0 = ub0k 1;k 1+ wTb0k;k+ rhb1k;k (u + wT + rh)b1k 1;k; 0 = ub1k 1;k 1+ wTb1k;k+ rhb0k;k (u + wT + rh)b0k;k 1;
0 = ub0k;k 1+ wTb0k+ 1;k+ rhb1k;k+ 1 (u + wT + rh)b1k;k; (A 23)
w hile for k = 0 we obtain
0 = wTb10;1+ wDb10;0+ rhb01;0 (u + wD)b00;0;
0 = wTb01;0+ wDb00;0+ rhb10;1 (u + wD)b10;0: (A 24)
Finally,from Eq.A 15 we have
0 = (u + wT)b1 (u + wT + rh)b0;
0 = (u + wT)b0 (u + wT + rh)b1: (A 25)
D ueto thesym m etry ofthesystem wecan concludethattheprobabilities b0k;k= b1k;kand b0k+ 1;k= b1k;k+ 1. T hen the solutionsofEqs.A 23 and A 24 can
be w ritten in the follow ing form ,
b0k;k= b1k;k= 12(1 q)q2k;
b0k+ 1;k= b1k;k+ 1= 12(1 q)q2k+ 1; (A 26)
w here k = 0;1;::,and
q= u
wT + rh
< 1: (A 27)
In addition,Eqs.A 25 have only a trivialsolution b0= b1= 0.R ecallthatb0 and b1 give the stationary-state probabilities ofthe polym er con gurations w ith allsubunits in AT P state,i.e.,it corresponds to the case ofvery large k. T he solution agrees w ith the results for b0k;k and b1k;k+ 1 at k ! 1 (see Eqs.A 26).
For q > 1 the system s of equations (A 23) and (A 24) have the trivial solutions,w ith allbik;m = 0 for nitek and m .Itm eansthatatthestationary conditions the polym er can only exist in the con gurations w ith very large num ber ofunhydrolyzed subunits and the size ofAT P-cap is in nite,w hile for q < 1 the size ofAT P-cap is always nite. T he case q = 1 isa boundary between two regim es. A t this condition there is a qualitative change in the dynam ic properties ofthe system .
To determ ine the coe cients aik;m and Tk;mi from Eq.A 22,the ansatz forthe functionsCk;mi is substituted into the asym ptotic expressions(A 16
-A 21),yielding for k 1,
0 = ua1k 1;k+ wTa1k;k+ 1+ rha0k+ 1;k (u + wT + rh)a0k;k; 0 = ua0k 1;k 1+ wTa0k;k+ rha1k;k (u + wT + rh)a1k 1;k; 0 = ua1k 1;k 1+ wTa1k;k+ rha0k;k (u + wT + rh)a0k;k 1;
0 = ua0k;k 1+ wTa0k+ 1;k+ rha1k;k+ 1 (u + wT + rh)a1k;k; (A 28)
A t the sam e tim e,for k = 0 we obtain
0 = wTa10;1+ wDa10;0+ rha01;0 (u + wD)a00;0;
0 = wTa01;0+ wDa00;0+ rha10;1 (u + wD)a10;0: (A 29)
T he coe cients Tk;mi satisfy the follow ing equations (for k 1),
a0k;k= uTk1 1;k+ wTTk;k+ 11 + rhTk+ 1;k0 (u + wT + rh)Tk;k0
+12[ub1k 1;k wTb1k;k+ 1]; (A 30)
a1k 1;k= uTk0 1;k 1+ wTTk;k0 + rhTk;k1 (u + wT + rh)Tk1 1;k
+12[ub0k 1;k 1 wTb0k;k]; (A 31)
a0k;k 1= uTk1 1;k 1+ wTTk;k1 + rhTk;k0 (u + wT + rh)Tk;k0 1
+12[ub1k 1;k 1 wTb1k;k]; (A 32)
a1k;k= uTk;k0 1+ wTTk+ 1;k0 + rhTk;k+ 11 (u + wT + rh)Tk;k1
+12[ub0k;k 1 wTb0k+ 1;k]: (A 33)
For k = 0 the expressions are given by
a00;0= wTT0;11 + wDT0;01 + rhT1;00 (u + wD)T0;00
1
2[wTb10;1+ wDb10;0]; (A 34)
a10;0= wTT10;0+ wDT00;0+ rhT01;1 (u + wD)T01;0 1
2[wTb01;0+ wDb00;0]: (A 35) C om paring Eqs.A 23 and A 24 w ith expressions A 28 and A 29,we con-clude that
aik;m = A bik;m; (i= 0;1); (A 36) w ith the constant A . T his constant can be calculated by sum m ing over the left and right sides in Eq. A 36 and recalling the norm alization condition (A 12). T he sum m ation over allak;iin Eqs.A 28 and A 29 produces
A =
+ 1P
k= 0
h
a0k;k+ a0k+ 1;k+ a1k;k+ 1+ a1k;k i
= 12 (u wT) (wD wT)(b00;0+ b10;0) : (A 37)
To determ ine the coe cients Tk;mi , we need to solve Eqs. A 30-A 35.
A gain, due to the sym m etry, we have Tk;k0 = Tk;k1 T2k, and Tk+ 1;k0 =
Tk;k+ 11 T2k+ 1 for allk. T he solutions for these equations are given by
w here k = 0;1;:::and T0 is an arbitrary constant.
It is now possible to calculate explicitly the m ean grow th velocity, V , and dispersion, D , at steady-state conditions. T he average length of the polym er is given by
< l(t)> = d
T hen,using Eq.A 36,we obtain for the velocity
V = lim A sim ilar approach can be used to derive the expression for dispersion.
W e start from
T hen,using the m aster equations (A 1 -A 6),it can be show n that
A lso,the follow ing equation can be derived using Eq.A 39,
t! 1lim
T he form alexpression for dispersion is given by
D = 1 2 lim
t! 1
d
dt < l(t)2> < l(t)>2 : (A 44) T hen,substituting into this expression Eqs.A 42 and A 43,we obtain
D = d2
Finally,aftersom ealgebraictransform ationsofEqs.A 37 and A 45,wederive the nalexpression for the grow th velocity,V and dispersion,D ,w hich are given in Eqs.1 and 2 in Section II.N ote thatthe constantT0cancels outin the nalequation.
T he m ean size ofAT P-cap can be calculated as
T he average relative uctuation in the size ofthe AT P-cap,by de nition,is given
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